The complexity of solving Weil restriction systems

TL;DR

This paper investigates the computational complexity of solving polynomial systems obtained through Weil restriction, providing upper bounds based on algebraic invariants, which aids in understanding their solvability via Groebner bases.

Contribution

It introduces upper bounds for the solving degree of Weil restriction systems, linking their complexity to algebraic properties of the original systems over field extensions.

Findings

Upper bounds for solving degree of Weil restriction systems

Complexity analysis based on algebraic invariants

Insights into solving polynomial systems over field extensions

Abstract

The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Groebner bases methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension $k\hookrightarrow K$, associates to a system $\mathcal{F}$ defined over $K$ a system $\mathrm{Weil}(\mathcal{F})$ defined over $k$, in such a way that the solutions of $\mathcal{F}$ over $K$ and those of $\mathrm{Weil}(\mathcal{F})$ over $k$ are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system $\mathrm{Weil}(\mathcal{F})$ obtained via Weil restriction in terms of algebraic invariants of the system $\mathcal{F}$.